Polyhedra not only occupy a prominent place in geometry, but also occur in the everyday life of every person. Not to mention artificially created household items in the form of various polygons, starting from a matchbox and ending with architectural elements, crystals in the form of a cube (salt), prisms (crystal), pyramids (scheelite), octahedron (diamond), etc. are also found in nature. . d.
The concept of a polyhedron, types of polyhedrons in geometry
Geometry as a science contains a section of stereometry that studies the characteristics and properties of three-dimensional figures. Geometric bodies whose sides in three-dimensional space are formed by limited planes (faces) are called polyhedra. Types of polyhedra have more than a dozen representatives that differ in the number and shape of faces.
Nevertheless, all polyhedrons have common properties:
- All of them have 3 integral components: a face (the surface of the polygon), a vertex (the angles formed at the junction of the faces), an edge (the side of the figure or a segment formed at the junction of two faces).
- Each edge of the polygon connects two, and only two faces, which are adjacent to each other.
- Bulge means that the body is completely located only on one side of the plane on which one of the faces lies. The rule applies to all faces of a polyhedron. Such geometric figures in stereometry are called the term convex polyhedra. The exception is stellate polyhedra, which are derivatives of regular polyhedral geometric bodies.
Polyhedra can be divided into:
- Types of convex polyhedra, consisting of the following classes: ordinary or classical (prism, pyramid, parallelepiped), regular (also called Platonic solids), semi-regular (second name - Archimedean bodies).
- Non-convex polyhedra (star-shaped).
Prism and its properties
Stereometry as a branch of geometry studies the properties of three-dimensional figures, the types of polyhedra (a prism among them). A prism is a geometric body that has two absolutely identical faces (they are also called bases) lying in parallel planes and the nth number of side faces in the form of parallelograms. In turn, the prism also has several varieties, including such types of polyhedra as:
- A parallelepiped - is formed if a parallelogram is at the base - a polygon with 2 pairs of equal opposite angles and two pairs of congruent opposite sides.
- The direct prism has ribs perpendicular to the base.
- An inclined prism is characterized by the presence of indirect angles (other than 90) between the faces and the base.
- A regular prism is characterized by bases in the form of a regular polygon with equal side faces.
The main properties of the prism:
- Congruent grounds.
- All edges of the prism are equal and parallel to each other.
- All side faces are in the form of a parallelogram.
Pyramid
A pyramid is a geometric body that consists of one base and of the nth number of triangular faces connecting at one point - the vertex. It should be noted that if the lateral faces of the pyramid are necessarily represented by triangles, then at the base there can be a triangular polygon, a quadrangle, and a pentagon, and so on to infinity. In this case, the name of the pyramid will correspond to the polygon at the base. For example, if a triangle lies at the base of the pyramid, it is a triangular pyramid , a quadrangle is a quadrangular, etc.
Pyramids are cone-shaped polyhedra. The types of polyhedra of this group, in addition to the above, also include the following representatives:
- A regular pyramid has a regular polygon at the base, and its height is projected to the center of the circle inscribed in the base or circumscribed around it.
- A rectangular pyramid is formed when one of the side edges intersects the base at a right angle. In this case, this edge is also rightly called the height of the pyramid.
Pyramid Properties:
- If all the side edges of the pyramid are congruent (of the same height), then they all intersect the base at the same angle, and around the base you can draw a circle with a center that coincides with the projection of the top of the pyramid.
- If the base of the pyramid is a regular polygon, then all the side edges are congruent, and the faces are isosceles triangles.
Regular polyhedron: types and properties of polyhedrons
In stereometry, a special place is occupied by geometric bodies with absolutely equal faces, at the vertices of which the same number of edges connects. These bodies are called Platonic solids, or regular polyhedra. Types of polyhedra with such properties have only five figures:
- Tetrahedron.
- Hexahedron
- Octahedron.
- Dodecahedron.
- Icosahedron
The correct polyhedrons owe their name to the ancient Greek philosopher Plato, who described these geometric bodies in his works and connected them with the natural elements: earth, water, fire, air. The fifth figure was awarded a resemblance to the structure of the Universe. In his opinion, the atoms of natural elements in shape resemble the types of regular polyhedra. Due to their most exciting property - symmetry, these geometric bodies were of great interest not only to ancient mathematicians and philosophers, but also to architects, artists and sculptors of all time. The presence of only 5 types of polyhedra with absolute symmetry was considered a fundamental find, they were even awarded a connection with the divine principle.
Hexahedron and its properties
In the form of a hexagon, Plato's successors assumed a similarity with the structure of the atoms of the earth. Of course, at present, this hypothesis is completely refuted, which, however, does not prevent figures from attracting the minds of famous figures with their aesthetics in modern times.
In geometry, a hexahedron, aka a cube, is considered a special case of a parallelepiped, which, in turn, is a kind of prism. Accordingly, the properties of the cube are associated with the properties of the prism with the only difference being that all the faces and angles of the cube are equal to each other. The following properties follow from this:
- All edges of the cube are congruent and lie in parallel planes with respect to each other.
- All faces are congruent squares (there are 6 in total in a cube), any of which can be taken as a basis.
- All interface angles are 90.
- An equal number of edges comes out from each vertex, namely 3.
- The cube has 9 axes of symmetry, which all intersect at the intersection of the diagonals of the hexahedron, called the center of symmetry.
Tetrahedron
A tetrahedron is a tetrahedron with equal faces in the shape of triangles, each of the vertices of which is a connection point of three faces.
Properties of a regular tetrahedron:
- All faces of a tetrahedron are equilateral triangles, which implies that all faces of a tetrahedron are congruent.
- Since the base is represented by a regular geometric figure, that is, it has equal sides, the faces of the tetrahedron converge at the same angle, that is, all angles are equal.
- The sum of the flat angles at each vertex is 180, since all the angles are equal, then any angle of the regular tetrahedron is 60.
- Each of the vertices is projected to the point of intersection of the heights of the opposite (orthocenter) face.
Octahedron and its properties
Describing the types of regular polyhedra, one cannot fail to note such an object as the octahedron, which can be visually represented as two quadrangular regular pyramids glued together by the bases.
Octahedron Properties:
- The name of the geometric body itself suggests the number of its faces. The octahedron consists of 8 congruent equilateral triangles, in each of the vertices of which an equal number of faces converges, namely 4.
- Since all faces of the octahedron are equal, its inter-facet angles are equal, each of which is 60, and the sum of the flat angles of any of the vertices is, therefore, 240.
Dodecahedron
If you imagine that all the faces of a geometric body are a regular pentagon, you get a dodecahedron - a figure of 12 polygons.
Dodecahedron Properties:
- At each vertex, three faces intersect.
- All faces are equal and have the same length of edges, as well as equal area.
- The dodecahedron has 15 axes and planes of symmetry, and any of them passes through the top of the face and the middle of the opposite edge.
Icosahedron
No less interesting than the dodecahedron, the icosahedron figure is a three-dimensional geometric body with 20 equal faces. Among the properties of a regular hexahedron are the following:
- All faces of the icosahedron are isosceles triangles.
- Five faces converge at each vertex of the polyhedron, and the sum of adjacent vertex angles is 300.
- The icosahedron has, like the dodecahedron, 15 axes and planes of symmetry passing through the midpoints of opposite faces.
Semi-Regular Polygons
In addition to Platonic solids, the group of convex polyhedra also includes Archimedean solids, which are truncated regular polyhedra. Types of polyhedra of this group have the following properties:
- Geometric bodies have pairwise equal faces of several types, for example, a truncated tetrahedron has 8 faces in the same way as a regular tetrahedron, but in the case of an Archimedean body, 4 faces will be triangular and 4 faces hexagonal.
- All angles of one vertex are congruent.
Star polyhedrons
Representatives of the voluminous types of geometric bodies are stellated polyhedra, the faces of which intersect with each other. They can be formed by the merger of two regular three-dimensional bodies or as a result of the continuation of their faces.
Thus, such stellate polyhedra are known as: stellate forms of the octahedron, dodecahedron, icosahedron, cuboctahedron, icosododecahedron.